Lehmann–Scheffé theorem

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In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation.[1] The theorem states that any estimator that is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers.[2][3]

If T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of τ(θ).

Statement

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Let   be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case)   where   is a parameter in the parameter space. Suppose   is a sufficient statistic for θ, and let   be a complete family. If   then   is the unique MVUE of θ.

Proof

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By the Rao–Blackwell theorem, if   is an unbiased estimator of θ then   defines an unbiased estimator of θ with the property that its variance is not greater than that of  .

Now we show that this function is unique. Suppose   is another candidate MVUE estimator of θ. Then again   defines an unbiased estimator of θ with the property that its variance is not greater than that of  . Then

 

Since   is a complete family

 

and therefore the function   is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that   is the MVUE.

Example for when using a non-complete minimal sufficient statistic

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An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is not complete, was provided by Galili and Meilijson in 2016.[4] Let   be a random sample from a scale-uniform distribution   with unknown mean   and known design parameter  . In the search for "best" possible unbiased estimators for  , it is natural to consider   as an initial (crude) unbiased estimator for   and then try to improve it. Since   is not a function of  , the minimal sufficient statistic for   (where   and  ), it may be improved using the Rao–Blackwell theorem as follows:

 

However, the following unbiased estimator can be shown to have lower variance:

 

And in fact, it could be even further improved when using the following estimator:

 

The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.[5]

See also

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References

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  1. ^ Casella, George (2001). Statistical Inference. Duxbury Press. p. 369. ISBN 978-0-534-24312-8.
  2. ^ Lehmann, E. L.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. I." Sankhyā. 10 (4): 305–340. doi:10.1007/978-1-4614-1412-4_23. JSTOR 25048038. MR 0039201.
  3. ^ Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II". Sankhyā. 15 (3): 219–236. doi:10.1007/978-1-4614-1412-4_24. JSTOR 25048243. MR 0072410.
  4. ^ Tal Galili; Isaac Meilijson (31 Mar 2016). "An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator". The American Statistician. 70 (1): 108–113. doi:10.1080/00031305.2015.1100683. PMC 4960505. PMID 27499547.
  5. ^ Taraldsen, Gunnar (2020). "Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comment". The American Statistician. 74 (3): 315. doi:10.1080/00031305.2020.1769727. S2CID 219493070.